Few body systems in a shell model approach

Few-Body Systems in a Shell-Model
Approach

Dissertation
zur
Erlangung des Doktorgrades (Dr. rer. nat.)
der
Mathematisch-Naturwissenschaftlichen Fakult¨at
der
Rheinischen Friedrich-Wilhelms-Universit¨at Bonn

vorgelegt von
Simon T¨olle
aus
Siegburg

Bonn 2013


Angefertigt mit der Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨at der Rheinischen Friedrich-Wilhelms-Universit¨at Bonn

Referent: Prof. Dr. Hans-Werner Hammer
Korreferent: PD. Dr. Bernard Ch. Metsch

Tag der Promotion: 10.02.2014
Erscheinungsjahr: 2014

II


III

Zusammenfassung
Im Rahmen dieser Arbeit werden zun¨achst Implementierungen zweier verschiedener Schalenmodelle zur Bestimmung von Bindungsenergien in bosonischen Mehrteilchensystemen vorgestellt und
verglichen.
Schwerpunktm¨aßig verwende ich das Schalenmodell zur Beschreibung von Bosonen mit Kontaktwechselwechselwirkungen, die in einem Oszillatorpotential eingesperrt sind, als auch f¨ur wechselwirkende 4 He-Atome und ihre Clusterbildung. Ausgiebig werden Abh¨angigkeiten der Resultate im
Schalenmodell von seiner Modellraumgr¨oße untersucht und M¨oglichkeiten gepr¨uft, eine schnellere Konvergenz zu erreichen; wie etwa ein Verschmieren der Kontaktkr¨afte sowie eine unit¨are
Transformation der Potentiale. Hierbei werden Systeme betrachtet, die maximal aus zw¨olf Bosonen bestehen.
Zus¨atzlich wird ein Verfahren zur Bestimmung von Streuobservablen anhand von Energiespektren
von Fermionen im harmonischen Oszillator vorgestellt und gepr¨uft. Schlussendlich werden anhand
der Abh¨angigkeit von Energiespektren von der Oszillatorbreite Position und Breite von Streuresonanzen extrahiert.
Teile dieser Arbeit sind zuvor in folgenden Artikeln ver¨offentlicht worden:
• S. T¨olle, H.-W. Hammer, and B. Ch. Metsch, Universal few-body physics in a harmonic trap,
C. R. Phys. 12, 59 (2011).
• S. T¨olle, H. W. Hammer, and B. Ch. Metsch, Convergence properties of the effective theory
for trapped bosons, J. Phys. G 40, 055004 (2013).

IV


Abstract
In this thesis, I introduce and compare an implementation of two different shell models for physical
systems consisting of multiple identical bosons.
In the main part, the shell model is used to study the energy spectra of bosons with contact interactions in a harmonic confinement as well as those of unconfined 4 He clusters. The convergence of the
shell-model results is investigated in detail as the size of the model space is increased. Furthermore,
possible improvements such as the smearing of contact interactions or a unitary transformation of
the potentials are utilised and assessed. Systems with up to twelve bosons are considered.
Moreover, I test a procedure to determine scattering observables from the energy spectra of fermions in a harmonic confinement. Finally, the position and width of resonances are extracted from
the dependence of the energy spectra on the oscillator length.
Some parts of this thesis have been previously published in following articles:

• S. T¨olle, H.-W. Hammer, and B. Ch. Metsch, Universal few-body physics in a harmonic trap,
C. R. Phys. 12, 59 (2011).
• S. T¨olle, H. W. Hammer, and B. Ch. Metsch, Convergence properties of the effective theory
for trapped bosons, J. Phys. G 40, 055004 (2013).

V


Contents
1

Introduction

2

Physical Background
2.1 Scattering Theory . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Differential Cross Section . . . . . . . . . . . . . .
2.1.2 Green’s Function . . . . . . . . . . . . . . . . . . .
2.1.3 Partial-Wave S-Matrix . . . . . . . . . . . . . . . .
2.2 Effective Theories . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Basic Concept . . . . . . . . . . . . . . . . . . . .
2.2.2 Effective Field Theory . . . . . . . . . . . . . . . .
2.2.3 Local Non-Relativistic EFT . . . . . . . . . . . . .
2.2.3.1 Two-Body Scattering . . . . . . . . . . .
2.2.3.2 Three-Body Scattering . . . . . . . . . . .
2.3 Efimov Effect . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Efimov Effect and local EFT . . . . . . . . . . . . .
2.3.2 Efimov Effect with External Confinement . . . . . .
2.4 Similarity Renormalisation Group . . . . . . . . . . . . . .

2.5 Experimental Techniques . . . . . . . . . . . . . . . . . . .
2.5.1 Study of Atoms with Resonant Interactions in Traps
2.5.1.1 Feshbach Resonances . . . . . . . . . . .
2.5.1.2 Traps and Cooling . . . . . . . . . . . .
2.5.2 Investigation of Helium Clusters by Diffraction . . .

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5
5
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7
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12
14
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19
19
19
20
21

Shell-Model Approach
3.1 J-Scheme Shell Model in Jacobi Coordinates . . . . .
3.1.1 Symmetric Basis . . . . . . . . . . . . . . . .

(A−1)→A
3.1.2 Explicit calculation of Csym
. . . . . . . .
3.1.3 Model Space and Elements of the Hamiltonian
3.1.4 Numerical Approach . . . . . . . . . . . . . .
3.2 M -Scheme Shell Model in One-Particle Coordinates .
3.2.1 Symmetric Basis . . . . . . . . . . . . . . . .
3.2.2 Model Space and Matrix Elements of H . . . .
3.2.2.1 Shift of Centre-of-Mass Excitations .
3.2.3 Numerical Procedure . . . . . . . . . . . . . .
3.3 Comparison of both Shell Models . . . . . . . . . . .

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23
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32
35
35
37

Few Bosons in Traps
4.1 Framework in the Scaling Limit . . . . . . . . . . . . . . . . . . . . . . . . . . .

39
39

3

4

1

VI

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4.2

4.3

4.4
5

6

7

Energy Spectra in the Scaling Limit . . . . . .
4.2.1 Three-Body Sector . . . . . . . . . . .
4.2.2 Four-Body Sector . . . . . . . . . . . .
4.2.3 Systems with more Bosons . . . . . . .
Smeared Contact Interaction . . . . . . . . . .
4.3.1 Matrix Elements and Renormalisation .
4.3.2 Running of Coupling Constants . . . .
4.3.3 Analysis of Uncertainties . . . . . . . .
4.3.4 Energy Spectra . . . . . . . . . . . . .

4.3.4.1 Three Identical Bosons . . .
4.3.4.2 Four Identical Bosons . . . .
4.3.4.3 Five and Six Identical Bosons
Conclusion . . . . . . . . . . . . . . . . . . .

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.

Clusters of Helium Atoms
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
5.2 LM2M2 Potential . . . . . . . . . . . . . . . . . . .
5.3 SRG-Evolved LM2M2 Interaction . . . . . . . . . .
5.4 Effective Pisa Potential . . . . . . . . . . . . . . . .
5.4.1 Soft Pisa Potential . . . . . . . . . . . . . .
5.4.2 Hard Pisa Potentials . . . . . . . . . . . . .
5.4.2.1 Unevolved Hard Pisa Potentials . .
5.4.2.2 SRG-Evolved Hard Pisa Potentials
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . .
Miscellanea
6.1 Scattering Observables from Energy Spectra . . .
6.1.1 Atom-Dimer Scattering . . . . . . . . . .

6.1.1.1 Atom-Dimer Scattering Length
6.1.1.2 Atom-Dimer Effective Range .
6.1.1.3 Conclusion . . . . . . . . . . .
6.1.2 Dimer-Dimer Scattering . . . . . . . . .
6.2 Description of Resonances with a Shell Model . .

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42
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59

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81

Conclusion and Outlook

85

A Jacobi Coordinates

88

B Talmi-Moshinsky Transformation

89

C Smeared Contact Interactions
C.1 Matrix Elements of Smeared Contact Interactions . . . . . . . . . . . . . . . . . .
C.2 Effective Range Expansion for Smeared Contact Interactions . . . . . . . . . . . .

90
90
91

D Dawson Integral

93

E Specification of the LM2M2 Potential

96

VII


VIII

Chapter 1
Introduction
Strongly correlated systems play an important role in several fields of physics, ranging from atomic
and nuclear to condensed matter physics. The description and understanding of such systems is
challenging, since they defy a treatment by perturbative methods. A new perspective is offered in
the framework of effective theories and especially of effective field theories (EFT). In particular,
systems with a large magnitude of the scattering lengths |a| will be at the focus of this thesis. Below,
I shall introduce the concept of the scattering length a, discuss the importance of large scattering
lengths a and the description of such a system. But first, I shall cover some relevant experimental
issues.
In atomic physics an active field of research concerns the so-called ”BEC-BCS crossover”. This
means the transition from the phase of a Bose-Einstein condensate (BEC) of weakly interacting
bosons, consisting of tightly bound fermions, to bosonic pairs of weakly interacting fermions,
called the cooper pairs, in the Bardeen-Cooper-Schrieffer (BCS) phase. The former phase belongs to small positive scattering lengths with the BEC-limit 1/a → +∞. In contrast, the latter
phase is characterised by a small negative scattering length with the BCS-limit 1/a → −∞. Consequently, the crossover happens in the vicinity of the resonance where the interaction leads to an
unnatural absolutely large scattering length 1/a ≈ ±0. After the discovery of high-temperature
superconductors in 1986 and the realisation that their phase seemed to be related to this crossover,
a lot of effort was made to examine the phenomenon in other experiments. In 1995, BEC’s could
finally be realised in gases of rubidium by Anderson et al. [1]. Great progress was made with the
realisation of a BEC in 6 Li and 40 K by various groups in 2003 [2–4]. These systems enabled a
deeper investigation of the crossover with the help of Feshbach resonances, since Feshbach resonances permit a continuous modification of the inter-particle interaction through external magnetic
fields and thus a tuning of the scattering length a. An extensive review of the research about the
BEC-BCS crossover is given in [5].
Strongly interacting systems with large scattering length occur also in nuclear physics. Prominent
examples are the proton-neutron system [6] and the scattering of α particles [7] as well. Furthermore, halo nuclei are at the focus of experimental research [8]. Along with large scattering lengths,
they are characterised by a small nucleon separation energy and a large radius, i.e. a long tail in
the nucleon density distribution. The main characteristic of halo nuclei is that the inner core is
surrounded by weakly bound nucleons. In nature, several halo nuclei could be identified: for example 11 Li, the Borromean nucleus 6 He and the most exotic nucleus 8 He with four weakly bound

neutrons.
1


2

Chapter 1. Introduction

A successful theoretical approach towards understanding the low energy physics for strongly correlated systems is the application of effective theories. They exploit a separation of scales in
systems in order to find the appropriate degrees of freedom and describe their behaviour in a
model-independent and systematically improvable way. At each order of the corresponding effective theory there is a fixed number of unknown effective parameters which have to be matched
to observables. In the context of quantum field theories the technique of effective theories is used in
a multitude of applications. A prominent example is chiral perturbation theory (ChPT), an effective
description of quantum-chromo-dynamics (QCD) at low energies [9]. Another example is the halo
EFT, which is successfully applied for halo nuclei mentioned above and is based on a dominant
large scattering length a [10].
In this thesis, I shall use the framework of the non-relativistic local EFT. Since I shall concentrate
on small momenta, a non-relativistic approximation is justified. In the case of non-relativistic field
theories, quantum field theory is equivalent to quantum mechanics; such field theories conserve the
particle number. Consequently, the principles of effective field theories (EFT) can be applied to
quantum mechanical problems, as pointed out by Lepage [11]. Hence, I shall work in a quantum
mechanical framework. Deeply connected to the non-relativistic EFT’s is the effective range expansion (ERE) in non-relativistic scattering theory [12]. The ERE is the low energy expansion in the
squared momentum k 2 of the scattering phase shift δ(k). The first and the second expansion parameter of the S-wave scattering phase shift are the negative inverse of the scattering length (−1/a)
and the effective range r0 , respectively. These parameters can serve as scattering observables to
determine the effective parameters in the EFT.
In the non-relativistic local EFT, the Hamiltonian is expressed as the integral of a Hamiltonian
density that depends on terms consisting of combinations of quantum fields ψ and their gradients at
the same point. The form of the interaction terms in the Hamiltonian are restricted by the principle,
that the EFT has to fulfil the same symmetries as the fundamental theory, such as Galilean symmetry
[6]. In the situation of a dominant scattering length a, the leading interaction term is the two-body

contact interaction without any range. In this case, the in principle highly complicated potentials
are then approximated by schematic contact potentials. Accordingly, observables depend only on
the scattering length in first order. This limit with vanishing effective range is called the scaling
limit. It can be applied to very different physical systems. Therefore regimes with unnaturally large
a are called universal. The theoretical interesting limit of a → ∞ is called the unitary limit.
In the three-body system, a new effect occurs in the vicinity of the unitary limit, which was predicted by Efimov in 1970 [13]. The Efimov effect signifies that in the universal regime there are
three-body bound states, so-called trimers, with binding energies which are approximately related
to the geometric series. In the unitary limit, there are infinitely many trimers with binding energies exactly related to the geometric series with an accumulation point at the 3-body scattering
threshold. The first experimental evidence for an Efimov trimer was provided in a trapped gas of
ultra-cold Cs atoms by its signature in the 3-body recombination rate [14]. Since this pioneering
experiment, there has been significant experimental progress in studying ultra-cold quantum gases
and in several experiments the Efimov effect could be detected [15]. So far these experiments were
carried out in a regime where the influence of the trap on the few-body spectra could be neglected.
However, the trap also offers new possibilities to modify the properties of few-body systems. In
particular, narrow confinements can lead to interesting new phenomena.
In the first part of this thesis, I shall focus on these effects. This work is partially an extension of
my diploma research topic [16]. For the sake of simplicity the confinement potential is idealised
by an isotropic harmonic oscillator potential (HOP). For such an harmonic confinement, the energy


3

spectrum for the two-body sector was determined in the scaling limit by Busch [17]. Furthermore,
the binding energies of three-body states could be found in the unitary limit [18]. The main observation is that there are two types of states: The first type includes states, which are completely
specified by the scattering length a. States, which belong to the second type, are called Efimov-like
and are fixed by the scattering length and an additional three-body parameter. For finite scattering
lengths and systems with more particles analytic solutions are unknown.
An established method to treat a confined strongly correlated few body system with spherical symmetry is the shell model. The basic idea is that the infinite-dimensional Hilbert space spanned by
(anti-)symmetric products of so-called single-particle wave functions is truncated e.g. by an energy
cutoff. Afterwards, a basis is chosen for this finite-dimensional model space. In this model space

the Schr¨odinger equation can be solved, since the Hamiltonian is just a finite matrix which can be
diagonalised numerically. There are several versions of shell model approaches which vary in details. I shall concentrate on shell models for bosons with a basis of symmetric products of harmonic
oscillator functions. Here, I work with the uncoupled oscillator basis in one-particle coordinates,
the so-called M -scheme, as well as with angular momenta coupled basis states expressed in relative
coordinates, the so-called J-scheme. Both methods have specific advantages and drawbacks which
are pointed out in section 3.3.
The second part is devoted to the description of 4 He clusters consisting of A atoms. The theoretical
and experimental investigation of atomic clusters is an important part of chemical physics. Helium
has two stable isotopes: the rare fermionic 3 He and the common bosonic 4 He. The latter has the
outstanding property that the Efimov effect can be observed directly because of the unnatural large
scattering length of 4 He atoms [6]. Furthermore, the understanding of 4 He clusters is the basis
to study properties of 4 He liquid droplets and the related phenomenon of super-fluidity of liquid
4
He [19]. Also the resonant absorption of nanosecond laser pulses in doped Helium nanodroplets
is an active area of research [20].
The existence of 4 He A-body clusters could be proved by diffraction experiments from a transmission grating [21]. Unfortunately, properties of the clusters cannot be measured in these experiments, e.g. even the binding energies are not directly observable in these experiments. Only in the
two body sector the binding energy of the two-body cluster, the dimer, can be deduced from its
size [22].
Various theoretical approaches have been used to investigate such systems and determine the binding energies. Moreover, several ab initio potentials for 4 He-4 He interaction are constructed within
different approaches. The potentials and these approaches are summarised in [23]. The binding
energies of the trimer ground and excited state are determined for a variety of these ab initio potentials. I shall concentrate on the so-called LM2M2 potential [23].
For few atoms the sizes and energies of A-body clusters have been calculated with Monte Carlo
methods and hyper-spherical adiabatic expansions. Up to the value A = 10 numerical results for
the ground and first excited states for the LM2M2 potential are presented in [24]. The challenging
part in the A-body calculations, as in nuclear physics for the nucleon-nucleon potential, is the treatment of the hard core repulsion of two 4 He atoms, which causes a coupling of low and high energy
physics. In order to solve the Hamiltonian numerically, some cutoff must be introduced. However,
the corresponding results contain large errors due to the coupling of the different energy scales. A
possible solution is to construct effective potentials and circumvent the hard core repulsion. For
instance, in [25] Gattobigio et al. propose a parametric interaction consisting of an attractive He-He
Gaussian potential with a contribution of a Gaussian-hyper-central three-body force, which reproduces the LM2M2 ground state trimer binding energy. Due to the research location of the majority

4

Chapter 1. Introduction

of the related research collaboration, I shall call this potential the Pisa potential. Gattobigio et al.
solved the Schr¨odinger equation with the Pisa potential in the hyper-spherical harmonic expansion
for up to six He-atoms and published the binding energies for the ground state and first excited
state [25].
There exists, however, a systematic procedure of the similarity renormalisation group (SRG) transformation to construct effective potentials based on unitary transformations. Numerical results
become more stable for SRG-transformed potentials at the expense of the introduction of effective
many body forces induced. In principle, these forces have to be considered for few body systems.
With my shell model methods for bosons I shall investigate the 4 He system for up to twelve particles. In cooperation with Prof. Forss´en from Gothenburg, I utilise the Pisa potential as well as the
LM2M2 potential as inter-particle potentials. For the purpose of better convergence, here indeed
the SRG evolution is exploited.
My thesis is organised as follows. In chapter 2, I outline the quantum mechanical scattering theory
and the basics of effective theories. Then the Efimov effect is elucidated and the SRG transformation is introduced. At the end of this chapter, relevant experimental techniques are mentioned,
which enable to observe the systems which I consider theoretically in this thesis. Subsequently,
I explain both the shell model approaches, which I used, in detail in chapter 3 and compare their
merits and demerits. In the following chapter 4, my results for few bosons in the scaling limit in
traps are presented. The calculations for atomic clusters of Helium atoms is the subject of chapter 5. In chapter 6, I collect alternative approaches and ideas. Finally, I summarise my results and
give an outlook of possible further studies in chapter 7.


Chapter 2
Physical Background
In this chapter I introduce the theoretical concepts and basic principles of experiments for the physical systems considered. At first, the basics of scattering theory are summarised in section 2.1. The
definition of differential cross sections, the connection to Green’s functions as well as the partialwave S-matrix are outlined. Afterwards, I give an introduction to effective theories in section 2.2
and explain the local non-relativistic effective field theory (EFT) which will be utilised for resonant

interactions. In section 2.3 the Efimov effect is elucidated with and without a confining trap in the
form of an oscillator potential. Subsequently, the similarity-renormalisation-group (SRG) transformation method is explained in section 2.4, as I need this technique to handle realistic potentials. At
the end, in section 2.5 I mention some experimental techniques for observing the physical systems
considered theoretically in my thesis.

2.1

Scattering Theory

Here, I present an overview on the quantum theory of non-relativistic, elastic scattering. It follows
the introduction to scattering theory in the textbook of Taylor [12].

2.1.1 Differential Cross Section
For the sake of simplicity, I describe the scattering of a projectile on an infinite-heavy target
described by a potential. The reformulation for two-particle scattering in relative coordinates is
straightforward.
The starting point is the time-dependent Schr¨odinger equation with the Hamiltonian H. The Hamilp2
tonian consists of the free part, i.e. the kinetic energy H0 = 2m
, and the time-independent potential
V . The formal solution of the initial value problem is given with the time evolution operator U (t)
as
i

∂
ψ(t) = H ψ(t) ,
∂t

U (t) ψ := e−iHt ψ(0) = ψ(t) .

(2.1)

In general, a scattering experiment is designed to start with a free incoming wave packet, the asymptotic state ψin (t) = U 0 (t) ψin before the actual scattering process, and to detect long time after
5


6

Chapter 2. Physical Background

the scattering (t → ∞) a free outcoming wave packet, the asymptotic state ψout (t) . Both packets
are asymptotes of the actual orbit U (t) ψ , i.e.
U (t) ψ −U 0 (t) ψin

t→−∞

−→ 0 ,

U (t) ψ −U 0 (t) ψout

t→+∞

−→ 0 .

(2.2)

The maps between the actual orbit and the asymptotes define the Møller wave operators Ω+ and
Ω− by
ψ = Ω+ ψin = lim U † (t)U 0 (t) ψin ,
t→−∞

ψ = Ω− ψout = lim U † (t)U 0 (t) ψout .
t→+∞

(2.3)

Thus, the whole scattering process from ψin to ψout is described by a combination of Møller
operators, which is called the S-matrix S. One finds
ψout = S ψin := Ω†− Ω+ ψin .

(2.4)

In momentum space equation (2.4) becomes
p ψout =

d3 q
p S q q ψin ,
(2π)3

(2.5)

with the S-matrix elements p S q . In the end, one is interested in the scattering part of the wave
function. Therefore, the trivial contribution is separated. The rest of the S-matrix defines the onshell T-matrix t(q ← p). Because of energy conservation a factor δ(Eq − Ep ) occurs and one
writes
p S q = (2π)3 δ (3) (q − p) − 2πi δ(Eq − Ep ) t(q ← p) ,

(2.6)

which defines t(q ← p). The observable in scattering experiments of spinless particles is the
differential cross section. The typical situation of scattering experiments is schematically depicted
in Figure 2.1.

dΩ

ψin,ρ (t)

1111
0000
p0
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111

ψin (t)

1111
0000
0000
1111
0000
1111
0000
1111
0000
1111

0000p0
1111

ρ
Target

Figure 2.1: Two wave packets ψin,0 (t) = ψin (t) and ψin,ρ (t) approach the target with their
identical mean momentum p0 and two different impact parameters |ρ|. The quantity w(dΩ ← ψin,ρ )
is the probability that ψin,ρ (t) is scattered into the solid angle dΩ.
The incident projectile approaches the target with the asymptote ψin . Now, the detector measures
the outgoing particle under a given solid angle dΩ. The experiment consists of a sequence of
independent collisions. In general the incoming wave packets will have different impact parameters
ρ. The cross section σ(dΩ ← ψin ) is the relation between the number of scattered particles in the
solid angle dΩ and all incoming particles with different displacements ρ. Thus, the cross section is


2.1. Scattering Theory

7

the integration over the impact parameter of the probability w(dΩ ← ψin,ρ ). Here, w(dΩ ← ψin,ρ )
is the probability that the projectile with ρ is scattered into the solid angle dΩ. The cross section
can be expressed with the outgoing asymptote ψout,ρ via
∞

d2 ρ

σ(dΩ ← ψin ) = dΩ

0

dp 2
p |ψout,ρ (p)|2 .
(2π)3

(2.7)

By means of equation (2.5), this yields
σ(dΩ ← ψin ) = dΩ

∞

2

dρ
0

dp 2
p
(2π)3

d3 q
p S q ψin,ρ (q)
(2π)3

2

.

(2.8)

In general, the incoming asymptote is designed to be peaked around a definite momentum p0 and
one measures approximately σ(dΩ ← p0 ). The combination of equations (2.6) and (2.7) then leads
to
m2
|t(p ← p0 )|2 dΩ ,
(2π)2
dσ
:=
(p ← p0 ) dΩ
dΩ

σ(dΩ ← p0 ) =

with the differential cross section
m
as f (p ← p0 ) := − 2π
t(p ← p0 ).

dσ
(p
dΩ

(2.9)
(2.10)

← p0 ). The scattering amplitude f (q ← p) is then defined

2.1.2 Green’s Function
Green’s functions are an important concept for scattering theory. The full Green’s function G(z)

and the free Green’s function G0 (z) are formally the resolvents of the Hamilton operator H =
H0 + V and of the free Hamiltonian H0 , respectively:
G(z) := (z − H)−1 ,

G0 (z) := (z − H 0 )−1 .

(2.11)

The Green’s function is analytic in the complex energy plane apart from specific points like bound
state energies or resonances, and it has a branch cut on the real axis from 0 to ∞. In scattering
theory another useful operator is the T-matrix defined as
T (z) = V + V G(z)V .

(2.12)

The integral equations, which relate the Green’s function G(z), G0 (z) and T (z), are the LippmannSchwinger equations for G(z) and for T (z):
G(z) = G0 (z) + G0 (z)V G(z) ,

T (z) = V + V G0 (z)T (z) .

(2.13)

2

p
Then, for z = limǫ↓0 ( 2m
+ iǫ) one finds

t(q ← p) = lim q T (p2 /(2m) + iǫ) p ,
ǫ↓0

(2.14)

with |q| = |p| for elastic scattering. Please note, that q T (z) p is more general than t(q ← p).
In the complex plane of z, poles in the T-matrix reflect important physical properties. We define
z = k 2 /(2m) and in the k-plane the poles in the T-matrix T (k 2 /(2m)) correspond to bound states,
if k lies on the positive imaginary axis. If k has an negative imaginary part and a non-vanishing real
part, these poles correspond to resonances. The poles on the negative imaginary axis are unphysical
virtual states.


8

Chapter 2. Physical Background

2.1.3 Partial-Wave S-Matrix
In the following, I shall focus on stationary scattering theory. For a stationary plane wave p the
scattered wave function is given by p+ := Ω+ p . For large distances from the scattering region
the scattered wave function has the following asymptotic behaviour
x p+

|x|=r→∞

−→

(2π)−3/2 eip·x + f (pˆ
x ← p)

eipr
.

r

(2.15)

In case of central forces, the S-matrix is diagonal in the angular momentum quantum number ℓ
as well as in the corresponding projection quantum number m and one finds for the scattering
amplitude a multipole expansion
f (q ← p) =

ℓ

(2ℓ + 1)fℓ (p)Pℓ (ˆ
q · pˆ = cos(θ)) ,

(2.16)

where qˆ := q/|q| and Pℓ are the Legendre polynomials [97]. Unitarity of the S-matrix guarantees
that the amplitude can be expressed in terms of the scattering phase δℓ (p) through
fℓ (p) =

1
.
p cot(δℓ (p)) − ip

(2.17)

For low energies the scattering phase behaves as δℓ ∼ p2ℓ+1 . Therefore, the s-wave scattering
(ℓ = 0) dominates the scattering in the vicinity of the threshold p = 0. The term p cot δ0 (p) can be
expanded in even powers of p. This is called the effective range expansion (ERE)
1 1

p cot(δ0 (k)) = − + r0 p2 + O(p4 ) .
a 2

(2.18)

For p ≈ 0 the scattering length a dominates and determines at leading order all scattering quantities.
Poles in the scattering amplitude for non-negative imaginary p lead to bound states. For the limit
a ≫ r0 > 0, there is a pole in the vicinity of p = +i/a; corresponding to a shallow bound state
at E ≈ − 1/(2ma2 ) ≈ 0. In general, the sign of a is crucial for the physical interpretation.
A negative a means that the scattering potential is attractive but too weak to build a bound state.
Positive scattering lengths are ambivalent: The potential can be either repulsive or attractive, when
a bound state could emerge.

2.2

Effective Theories

Effective theories have proved to be very useful in a vast variety of physical systems to describe
low energy properties. The application of the concept of effective theories in the quantum field
theory framework is called effective field theory (EFT). In section 2.2.1 I give an overview of the
basic concepts of effective theories. Subsequently, I consider EFT’s and the so-called local EFT
for non-relativistic particles with short range interactions. I orient myself on the treatments given
in [6, 26].

2.2.1 Basic Concept
An effective treatment is based on a separation of scales for a specific physical system. Separation
of scales means that the system has at least two scales with the property Λlow ≪ Λhigh . In the


2.2. Effective Theories

9

dimensionless small quantity Λlow /Λhigh ≪ 1 then a perturbative expansion can be performed. The
expansion up to different orders then defines a tower of effective theories.
A popular simple example is an approximation of Newton’s gravitational law, which reads VNewton =
−GM m/(R + h) for an object with mass m in a height h above the earth which has the mass M
(eff)
and radius R, by a constant earth acceleration g, which yields VNewton = +mgh. The high scale is
the radius Λhigh = R and the low scale is the height Λlow = h. Newton’s potential can be expanded
in Λlow /Λhigh = h/R and one has
VNewton = −

GM
h
1
m 1− +
R
R 2

h
R

2

+O

h
R

3

.

(2.19)

The first term is just a constant and irrelevant, since potentials can be measured only relatively
and not absolutely. The leading order in the effective theory is exactly mgh with identification
of GM/R2 = g as the coupling constant. At next to leading order a second coupling constant
g2 = −GM/(2R3 ) appears in the term g2 mh2 .
In spite of the simplicity of this example, several characteristic features of effective theories are
evident: The accuracy of the theory can be improved systematically with each order. The small
expansion parameter establishes a so-called power-counting scheme to sort the expansion terms by
importance. Coupling constants absorb the physics at the high scale and finally, the theory predicts
the scale of its own collapse at h ≈ R. In general the coupling constants cannot be determined by
the fundamental theory. Either the fundamental theory is unknown or the fundamental theory is too
complex to calculate the coupling constants explicitly. In these cases, the coupling constants are to
be determined by experimental data.

2.2.2 Effective Field Theory
Historically, EFT’s have been formulated in the field of nuclear physics. This was encouraged by
the endeavour to overcome model dependent descriptions of hadronic and nuclear properties and to
find a model-independent approach for the strong interaction with QCD as the fundamental theory.
Nowadays, two effective theories are used for strong interactions: chiral effective field theory
(χEFT) [9] and the pionless EFT (π/EFT) [27]. The former bases on the approximate chiral symmetry of the QCD Lagrangian, i.e. the invariance under the separate transformations of left-handed
and right-handed fields with the group SU(3)L ×SU(3)R . However, the chiral symmetry is hidden
due to spontaneous breaking of this group to SU(3)V . According to the Goldstone theorem, this
spontaneous breaking induces eight mass-less Goldstone bosons. Consequently, the eight lightest
mesons in the spectrum, the pions, kaons and η, are identified with the Goldstone bosons as the
explicit dynamical degrees of freedom. Since the symmetry is broken explicitly by the small and

various quark masses, the Goldstone bosons obtain finite masses.
Thus, the χEFT is an expansion around the chiral limit and has two high scales Λχhigh . The first one
is the so-called chiral symmetry breaking scale ΛCSB = 4πfπ where fπ is the pion decay constant.
The second one is the mass mρ of the lightest vector meson ρ, which is integrated out and is not a
dynamical degree of freedom anymore. The low scales Λχlow are the momentum p of the considered
process and the masses of the dynamical degrees of freedom mπ,κ,η . Consequently, the expansion
is in powers of the small parameters Λχlow /Λχhigh , where
Λχlow ∈ {mπ , mκ , mη , p}

and

Λχhigh ∈ {4πfπ , mρ } .

(2.20)


10

Chapter 2. Physical Background

The pionless EFT (π/EFT) is used for nucleon-nucleon reactions for momenta p ≪ mπ . In χEFT the
nucleon interactions are strong and lead to non-perturbative phenomena. Thus, the effects cannot
be treated in the normal power-counting scheme of χEFT. The reason behind this is the unnatural
large scattering length a ≫ 1/Λχhigh of the nucleon-nucleon scattering and the associated shallow
1
bound states at the vicinity of E = 2µa
2 with the reduced mass µ. In the next section the basics of
the π/EFT are considered.

2.2.3 Local Non-Relativistic EFT

The local non-relativistic EFT relies on the scattering length being large and is independent of the
mechanism responsible for this. This is summarised in the term universality. This theory can be
applied in nuclear physics, known as π/EFT, in atomic as well as particle physics.
The starting points of EFT’s are the Lagrangian densities. In local quantum field theories the
Lagrangian density is constructed with terms consisting of a combination of quantum fields ψ and
their gradients at the same point only. Accordingly to the symmetry principle [28], e.g. all terms
which fulfil Galilean symmetry, are included. Furthermore, terms which differ only by integration
by parts are equivalent, because the difference is just a boundary term. Since one is interested in
small momenta, terms with higher derivatives are suppressed. Consequently, at leading order the
Lagrangian is given by
L = ψ † (x, t) i

∂
1
g2 †
+
∆x ψ(x, t) −
ψ (x, t)ψ(x, t)
∂t 2m
4

2

−

g3 †
ψ (x, t)ψ(x, t)
36

3

+ ··· .

(2.21)

The ellipses indicate the terms in higher power-counting order. The leading order corresponds to
the limit of zero range interactions, the so-called scaling limit. It is equivalent to the truncation of
the ERE after the effective range term 1/a, which is justified for 1/k ≥ a ≫ r0 .
This quantum field theory is equivalent to a quantum mechanical description because no antiparticles terms are present and sectors with different particle numbers decouple in the Fock space.
In short, the quantum fields ψ † (x, t) and ψ(x, t) create a particle at position x at time t or destroy a
particle, respectively. For bosons, the fields fulfil the commutator relation
ψ † (x, t), ψ(y, t)

−

= δ (3) (x − y) ,

(2.22)

and all other commutator relations vanish identically. With the relation to the Hamiltonian density
∂L ˙
ψ−L,
∂ ψ˙
g2 †
−∆
ψ(x, t) +
ψ (x, t)ψ(x, t)
= ψ † (x, t)
2m
4

H(x, t) =

(2.23)
2

+

g3 †
ψ (x, t)ψ(x, t)
36

3

+ ··· ,

(2.24)

the quantum field formulation can be rewritten in the quantum mechanical formulation with the
Hamiltonian H. As an example I consider the 2-body sector where the three-body term in the


2.2. Effective Theories

11

Lagrangian density is then irrelevant. After normal ordering, denoted by two colons, one finds
x, y
=

d3 z : H(z, t) : φ1 , φ2
d3 zd3 vd3 w x, y : ψ † (z, t)

−1
g2 †
∆z ψ(z, t) +
ψ (z, t)ψ(z, t)
2m
4

2

: v, w

v, w φ1 , φ2 ,
1
g2
1
∆x −
∆y + δ (3) (x − y) φ1 (x, t)φ2 (y, t) + φ1 (y, t)φ2 (x, t) ,
∝ −
2m
2m
2
= H φ1 (x, t)φ2 (y, t) + φ1 (y, t)φ2 (x, t) .

2.2.3.1

(2.25)
(2.26)

(2.27)

Two-Body Scattering

In this section I investigate the scattering of two identical bosons with the EFT described above. In
order to calculate the scattering amplitude from the Lagrangian in the scaling limit, the Feynman
rules read as follows: The Feynman propagator for a particle of mass m with energy k0 and momentum k is given by i/(k0 − k 2 /(2m) + iǫ). The only vertex contribution is a constant −ig2 . Note that
g2 is not small in general. Thus, the scattering amplitude cannot be determined perturbatively with
the Dyson series: a re-summation of the loop-diagram contributions has to be performed. Finally,
this yields the Lippmann-Schwinger equation for the T -matrix, see equation (2.13). At on-shell
energies the T -matrix coincides with the scattering amplitude, see equation (2.14).
In Figure 2.2 the equations for the scattering amplitude are depicted diagrammatically. It describes
scattering of two identical bosons with reduced mass µ = m/2, relative energy E = k 2 /(2µ) =
k 2 /m and momentum k. Thus, in the centre-of-mass frame for |k ′ | = |k| the integral equation
diagram reads
k ′ T (E) k
i
= +g2 + g22
2

d3 q
(2π)3

1
dq0
1
2
q
(2π) q0 −
+ iǫ E − q0 −

2m

q2
2m

+ iǫ

q T (E) k .

(2.28)

Because of the contact interaction, the T-matrix simplifies and depends only on the energy E and
is independent of the direction kˆ′ . At on-shell energies, E = |k|2 /(2µ), the T-matrix is related to
the scattering amplitude as follows:
A2 (E) := − lim k ′ T (E + iǫ) k = −t(k ′ ← k) = +2
ǫ↓0

2π
f (k ′ ← k) .
µ

(2.29)

Due to the identity of the bosons, there is an additional factor of two.
Note that the integral over q in equation (2.28) diverges. This signifies that the Lagrangian is
ill-defined as such and has to be regularised. Likewise, the corresponding Hamiltonian is not selfadjoint because of the contact interaction in the form of the term with the δ-distribution. A possible
regularisation is a momentum cutoff Λ, i.e. the integration is substituted by q≤Λ d3 q. Thereafter,
the integral become finite but cutoff-dependent with the result
A2 (E) = −g2 1 +

mg2
π√
−mE − iǫ
Λ−
2
4π
2

−1

.

(2.30)


12

Chapter 2. Physical Background

=

+

=

+

+···

+

Figure 2.2: Lippmann-Schwinger equation for the scattering of particles in leading order of the
local non-relativistic EFT.
In order to predict observables, the coupling constant must be renormalised and matched for a fixed
Λ to an observable with the result that g2 becomes a function of Λ, the running coupling constant.
Explicitly, g2 is renormalised by the constraint that the scattering length a is fixed:
µ
A2 (0) .
E→0 4π

!

−a = lim f (k ′ ← k) = lim
k→0

(2.31)

Then, the renormalised scattering amplitude becomes independent of the ultraviolet cutoff and one
finds
A2 (E + iǫ) =

4π
µ −1/a +

1
−(2µ)E − iǫ

.

(2.32)

1
The pole in the amplitude at ED = − 2µa
2 indicates the universal two-body bound, already mentioned in section 2.1.3.

At the end of this section, I would like to stress that the complete Λ-independence in the amplitude
is accidental. In general, one expects only a suppressed Λ-dependence with E/Λ2 . But, in the
scaling limit, the cutoff can be increased without bounds and at all energies the system can be
treated in the EFT. Note however, that in real physical systems a natural cutoff is usually given by
the inverse of the effective range 1/r0 . In short, the EFT is appropriate only for physical properties
at energy scales of E < 1/r02 .
2.2.3.2

Three-Body Scattering

The Lagrangian for the three-body sector now contains a three-body term. The description of threeboson scattering within the EFT in the scaling limit is rather intricate. In the centre-of-mass frame
the 6-point Green’s function in momentum space 0 T (ψψψψ † ψ † ψ † ) 0 depends on 4 momentum
vectors and 5 off-shell energies and such an solution of a integral equation in many variables is
highly complicated.
At this point, one introduces an auxiliary dimer field. The new quantum field d, the dimer field, is
a composite of two ψ’s. The 6-point Green’s function then reduces to a 4-point Greens function
A3 = 0 T (dψd† ψ † ) 0 . A new Lagrangian, which involves the field d explicitly and is equivalent
to the former Lagrangian in leading order, can be constructed and is written as
L3 = ψ † i

∂
∆
g2
g2 †
g3

+
ψ + d† d −
d ψψ + ψ † ψ † d − d† dψ † ψ .
∂t 2m
4
4
36

(2.33)


2.2. Effective Theories

13

Based on this Lagrangian, Figure 2.3 depicts in terms of Feynman diagrams the scattering equation
for three bosons, in the case of elastic boson-dimer scattering.

+

+

=

+

Figure 2.3: Three-body scattering rewritten in dimer-boson scattering.
The corresponding Feynman rules for the Lagrangian in equation (2.33) are collected in Figure 2.4.
Note that, the Feynman propagator of the field d is naively just a constant, but the dimer can be split
into two bosons. Thus, the full propagator is a sum over all loop diagrams which leads to the same

integral equation as in the 2-body scattering apart from constants (see equation (2.28)). At the end,
one has for the full dimer propagator with energy P0 and momentum P
iD(P0 , P ) = i

32π 1
−
mg22 a

−mP0 + P 2 /4 − iǫ

i
k0 −k 2 /2+iǫ

.

−i g22

4i
g2

=

−1

(2.34)

g3
−i 36

+

Figure 2.4: Feynman rules for Lagrangian in equation (2.33).
Using the Feynman rules with projection onto S-waves and with the help of the residue theorem,
one finds in the centre-of-mass frame for the dimer-boson amplitude
H(Λ)
p2 + pk + k 2 − E − iǫ
16π 1
ln 2
+
2
a 2pk
p − pk + k − E − iǫ
Λ2
4 Λ
AS (q, k; E)
1
H(Λ)
p2 + pq + q 2 − E − iǫ
+
dq q 2
ln 2
+
. (2.35)
2
2
π 0
2pq
p − pq + q − E − iǫ
Λ
− a1 + 34 q 2 − E − iǫ

AS (p, k; E) =

Here, p and k denote the absolute value of the incoming and the outcoming momenta, respectively
and E is the energy. Since the original integral is again divergent, the integral has been regularised
with the cutoff Λ. In order to match to observables and to compensate the Λ-dependence, the
coupling constants must be renormalised. The cutoff dependence of g3 is described in the function
H(Λ)/Λ2 := −g3 /(9g22 m). It can be shown, that H(Λ) is a periodic function related to a UV
renormalisation group limit cycle. H can be parametrised as
H(Λ) =

cos s0 ln
cos s0 ln

Λ
Λ∗
Λ
Λ∗

+ arctan s0
− arctan s0

.

(2.36)


14

Chapter 2. Physical Background

The scaling-violation parameter Λ∗ is related to an observable in the three-body sector up to a
multiplicative factor of exp(nπ/s0 ) with n ∈ N, where s0 is an universal constant. For identical
bosons one finds s0 ≈ 1.00624.
We will see that the integral equation (2.35) yields roughly geometrically distributed three-body
bound state energies, even for negative scattering lengths, where no two-body bound state exists.
The existence of these bound states is called the Efimov effect and it is discussed in the next section 2.3 in more detail.

2.3

Efimov Effect

As mentioned in [29], Efimov published his studies about the three-nucleon system interacting
through short-ranged interactions with the natural length scale l in 1970 [13]. For interactions
with large scattering lengths a ≫ l, he focused on the low-energy behaviour, i.e. E ≪ 2 /(ml2 ).
In doing so, he discovered a sequence of roughly geometrically distributed bound state energies
between 2 /(ml)2 and 2 /(ma)2 . If the scattering length is increased, new bound states appear in
the spectrum. For the unitary limit (a → ±∞), the bound state energies are exactly geometrically
distributed with an accumulation point at the threshold E = 0. Afterwards, the Efimov effect was
formally proved by Amado and Noble in the following two years [30,31]. Within the local EFT we
will see that the Efimov effect is observed in the three-body scattering in the modern language of
quantum field theory and that it is a manifestation of an UV renormalisation group limit cycle.
More than 30 years after the prediction of Efimov trimers, the first experimental evidence for these
were found in a trapped gas of ultra-cold Cs atoms [14]. In this experiment, signatures in the
three-body recombination rate for negative scattering lengths indicate the existence of trimer states.
Their dependence on the scattering length was studied by tuning the scattering length by Feshbach
resonances (see section 2.5.1.1). Since this pioneering experiment, a lot of progress was made and
Efimov physics was observed in several experiments.
Evidence for Efimov trimers in 3-body recombination was also obtained in a balanced mixture of
atoms in three different hyper-fine states of 6 Li [33, 34], in a mixture of Potassium and Rubidium

atoms [35], and in an ultra-cold gas of 7 Li atoms [36]. In another experiment with Potassium atoms
[37], two bound trimers were observed with energies compatible with the geometric prediction for
Efimov states. Efimov states can also be observed as resonances in atom-dimer scattering. Such
resonances have been seen with atom-dimer mixtures of Cs atoms [38] and of 6 Li atoms [39, 40].
The first direct observation of Efimov trimers of 6 Li atoms created by radio frequency association
was recently reported by the Heidelberg group [41].
These experiments were carried out in a regime where the influence of the trap on the system could
be neglected. However, it is to be expected that with experimental advances the trap frequencies can
be increased and the trap itself could be used to modify and study internal properties of few-body
systems.

2.3.1 Efimov Effect and local EFT
Within the local EFT the Efimov effect can be studied via the scattering amplitude of the three-body
scattering. It turns out to be automatically related to the renormalisation group concept [6]. As in
the two-body sector, poles in the scattering amplitude for negative energies signify the existence of a


2.3. Efimov Effect

15
(n)

bound state with binding energy ET . In order to extract the poles from the integral equation (2.35)
one exploits that the amplitude factorises in the vicinity of a pole:
AS (p, q; E) →

B (n) (p)B (n) (q)
E+

(n)

ET

(n)

as E → − ET .

(2.37)

Then, combining equations (2.35) and (2.37), the bound-state integral equation for the amplitude
of Efimov states is found:
B

(n)

q

2

4
(p) =
π

∞

dq

0

1
p2 + pq + q 2 − E − iǫ

H(Λ)
ln 2
+
2
2pq
p − pq + q − E − iǫ
Λ2

−1/a +

3 2
q − E − iǫ
4

−1

B (n) (q) . (2.38)

The values for E for which this homogeneous equation has solutions are the binding energies
(n)
(−ET ) of the Efimov states. In fact the spectrum depends on two parameters determined by two
observables. The first one is the scattering length a and the second one is the parameter Λ∗ in the
function H(Λ) which is matched to reproduce some binding energy of a single Efimov trimer in the
spectrum. The rest of the spectrum is then independent of the arbitrary cutoff Λ. It can be chosen as
˜ of the periodic function H(Λ). Λ
˜ is fixed only up to the multiplicative factors of (emπ/s0 )
a root Λ
˜ all Efimov states up to around the binding energy |E n | ≤ Λ
˜2
with integer m. For a specific cutoff Λ,

T
can then be calculated with equation (2.38) numerically. In order to compute more deeper bound
˜ of the function H(Λ).
Efimov states, the cutoff has to be increased, i.e. one chooses a larger root Λ
The spectrum for a fixed Λ∗ is shown schematically as a function of 1/a in Figure 2.5.
sgn(E)

T

T

T

T
˜1
−Λ

|E|
1/a

D
˜2
−Λ

Figure 2.5: Efimov states T for a specific Λ∗ as a function of 1/a with the dimer threshold D. The
˜ 1 the deeper
trimers are not drawn to scale. In fact, they scale with a factor of eπ/s0 ≈ 22.7. With Λ
˜ 1 can not be found. Thus, a larger cutoff Λ
˜ 2 is necessary to determine the
Efimov states below Λ

next Efimov state.
Note that the EFT describes systems in the scaling limit. The spectrum of a real physical system
is restricted from below since all interactions have a finite range and therefore a natural length l.
Exactly this additional scale restricts the scope of the theory. Hence, only Efimov trimers with
energies ET ≪ /(ml2 ) are described correctly. Physical properties at higher binding energies
depend on the details of the interaction beyond the EFT.


16

Chapter 2. Physical Background

2.3.2 Efimov Effect with External Confinement
Before reviewing the solution of the three-body problem in a harmonic trap, it is worthwhile to
inspect the confined two-body problem in the scaling limit.
I consider a two-body system confined by an isotropic harmonic oscillator. For simplicity, it is
/(mω) with the mass m and the fresensible to express all lengths in the oscillator length b :=
(b)
quency ω. The dimensionless Jacobi coordinates (see appendix A) are defined by si := si /b. The
(b)
(b)
contact interaction is parametrised with the regularised δ-distribution δ (3) (s1 ) ∂(b) s1 . The cou∂s1

pling constant is related to the scattering length a, see [16]. Thus, the corresponding Hamiltonian
in Jacobi coordinates reads
√ a
1
1 (b)
∂ (b)
(b)

H = ω − ∆s(b) + |s1 |2 + 2π δ (3) (s1 ) (b) s1 .
(2.39)
2 1
2
b
∂s1
The spectrum of this Hamiltonian has been determined by Busch et al. [17]. Only the spectrum for
vanishing relative angular momentum is modified by the contact term and the energies E are given
by solutions of
b √ Γ − 2Eω +
= 2
a
Γ − 2Eω +

3
4
1
4

.

(2.40)

E [ ω]

Therefore, the spectrum is completely specified by the scattering length a. In principle, the scattering length could be extracted from a measurement of the energy spectrum. Note that a specification
of the scattering length a is equivalent to specifying any energy of the spectrum. Both procedures
determine the whole spectrum and both quantities can be used as an observable to renormalise the
coupling constant in the effective theory.
8

7
6
5
4
3
2
1
0
-1
-2
-10

-5

0
1/a [1/b]

5

10

Figure 2.6: Spectrum of the two-body sector in the harmonic confinement as a function of 1/a [17].
The dots indicate the spectrum in the unitary limit b/a → 0.
The spectrum is depicted in Figure 2.6. For b/a → −∞ the result is the spectrum of the oscillator
without any contact interaction. This spectrum is lowered by ∆E = 1 ω in the unitary limit


2.4. Similarity Renormalisation Group

17

b/a → 0. For the limit b/a → +∞ the states are again lowered by ∆E = 1 ω apart from the
ground state, of which the energy diverges to −∞. Then again the oscillator spectrum is found
with an additional infinitely deeply bound ground state.
For an axial-symmetric or an anisotropic harmonic oscillator potential the two-body energy spectra
were derived analytically by Idziaszek et al. [42] and by Liang et al. [43], respectively.
The three-body sector is more complicated and the Hamiltonian cannot be solved in general. But
for the unitary limit, solutions are published for bosons by Jonsell et al. [44] and for bosons and
fermions by Werner et al. [18].
The most noteworthy point is that for the bosons there are two types of energy states for vanishing
relative angular momentum l = 0. On the one hand, there are states for |a|/b → ∞ with energies
En,q = ω(s0,n + 1 + 2q) ,

(2.41)

with a non-negative integer q and the positive real, not integer solutions s0,n of the transcendental
equation
+s0,n cos

π
8
π
s0,n = √ sin
s0,n
2
6
3

.

(2.42)

On the other hand, states of the second type are called Efimov-like. They belong to the single
imaginary solution s0,0 ≈ 1.0062 i of the transcendental equation. As for Efimov states, energies
of these states depend on a three-body parameter Rt in addition to the scattering length a/b → ∞.
The energies of Efimov-like states are the solutions of the following equation:
arg Γ

1 + s0,0 − E/( ω)
2

= −|s0,0 | ln(Rt /b) + arg Γ s0,0 + 1

mod π .

(2.43)

The spectrum of Efimov-like states is bounded neither from below nor from above. The scaling
with the factor of about 22.7, known from the unconfined three-body system, can be observed for
adjacent, large negative energies En , En+1 :
En
∼ (22.7)2 .
En+1

2.4

(2.44)

Similarity Renormalisation Group

In 1990’s, Glazek and Wilson [45, 46] as well as Wegner [47] developed independently the similarity renormalisation group approach (SRG). The first application to the nuclear many-body problem
was published in 2007 by Bogner et al. [48]. I will exploit the SRG-transformation for the few-body
sector of 4 He-atoms treated with the realistic ab-initio LM2M2 potential in section 5.
The motivation for the SRG approach is the strong intertwinement of high- and low-energy physics
for strong short-ranged potentials in particular for potentials with a hard core. Due to this, the
spectrum in shell-model calculations shows a strong cutoff dependence. Accordingly, only results
with large cutoffs are reliable, but at the same time, the determination is then very time-consuming
and elaborate. A possible expedient is the SRG approach which decreases the correlations at the
expense of introducing more-particle interactions.